In a large, heavy-bottomed saucepan, bring sugar, evaporated milk, and heavy cream to a boil over medium heat. Add corn syrup and continue cooking until mixture reaches 220 degrees F.
Add butter and vanilla; continue cooking, stirring constantly, until caramel reaches 225 degrees F. Remove from heat, stir in salt, and carefully pour into a lightly oiled 9" x 13" baking dish (parchment paper in the bottom of the pan is helpful).
Let cool at least 1 hour in the refrigerator. Invert onto a cutting board covered with waxed or parchment paper and cut into 1-inch pieces with a sharp, lightly oiled knife. After cutting, sprinkle each piece with a few flecks of coarse sea salt.
Caramels can be wrapped in waxed or parchment paper or cellophane. (To serve candy-store style, wrap individual pieces in waxed or parchment paper and secure the ends with a simple twist.) Store airtight at cool room temperature (around 65 degrees F) or in the refrigerator; will keep up to 3 weeks.